5.7: Slope
Have you ever thought of slope connected with books and reading? I know it sounds interesting. Take a look at this dilemma.
Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.
What is their rate? If you were to draw the rate of change on a graph comparing books to weeks, what would the graph look like?
To understand this problem, you have to understand slope and graphs. All of this will be shown in this Concept. At the end, you will understand how to show these values on a graph.
Guidance
Have you ever been skiing? Even if you haven't, you may know a little bit about what it might be like to learn to ski.
When someone first learns to ski, he or she usually starts on a slope that is not very steep. Sometimes that slope is called a beginners' slope. After mastering the basic skills for skiing, a person may begin to try slopes that are steeper and more challenging.
In mathematics, the term slope has a different meaning. In mathematics, the slope of a line describes the steepness of a line. However, thinking of a ski slope can help you remember that the slope of a line tells how steep it is.
It is helpful to think of the slope of the line as the “rise-over-run.” That is, the slope is the ratio of the vertical (up and down) rise of a line to its horizontal (left to right) run.
To help us understand this ratio, let's look at line \begin{align*}AB\end{align*} on the coordinate plane below.
Imagine placing your finger on point \begin{align*}A\end{align*}. To move from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*}, your finger would need to move 5 units up and then move 6 units to the right. That is because the line has a rise of 5 units up and a run of 6 units to the right.
So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{5}{6}\end{align*}.
The slope of line \begin{align*}AB\end{align*} is \begin{align*}\frac{5}{6}\end{align*}.
Line \begin{align*}AB\end{align*} has a slope of \begin{align*}\frac{5}{6}\end{align*}, which is a positive slope.
Notice that line \begin{align*}AB\end{align*} slants up from left to right!!
Knowing some basic information about the slope of a line can tell you about its slant.
- A line that slants up from left to right has a positive slope.
- A line that slants down from left to right has a negative slope.
Determine if the slope of each line shown below is positive or negative.
a.
b.
Consider the line in \begin{align*}a\end{align*}.
The line slants down from left to right, so its slope is negative.
Consider the line in \begin{align*}b\end{align*}.
The line slants up from left to right, so its slope is positive.
You should also know about the slopes of horizontal and vertical lines.
- A horizontal line has a run, but does not have a rise.
\begin{align*}\text{slope} = \frac{rise}{run} = \frac{0}{n} = 0.\end{align*}
So, the slope of a horizontal line is zero.
- A vertical line has a rise, but does not have a run.
\begin{align*}\text{slope} = \frac{rise}{run} = \frac{n}{0} = undefined.\end{align*}
Any fraction with a zero in the denominator is undefined. So, the slope of a vertical line is undefined.
Identify the slope of each line shown below.
a.
b.
Consider the line in \begin{align*}a\end{align*}.
The line is vertical, so its slope is undefined.
Consider the line in \begin{align*}b\end{align*}.
The line is horizontal, so its slope is zero.
Now it is your turn to think about slope. Answer the following questions.
Example A
What is the slope of a horizontal line?
Solution: 0
Example B
What is the slope of a vertical line?
Solution: Undefined
Example C
What is the slope of a line that goes up from left to right?
Solution: Positive
Here is the original problem once again.
Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.
What is their rate? If you were to draw the rate of change on a graph comparing books to weeks, what would the graph look like?
If you read the problem carefully you’ll see that the rate of the students is one book per week.
Let’s list their weeks and books in a table.
week | books |
---|---|
1 | 1 |
2 | 1 |
3 | 1 |
4 | 1 |
5 | 1 |
6 | 1 |
7 | 1 |
8 | 1 |
9 | 1 |
10 | 1 |
Now, let’s create a graph to show these results.
Vocabulary
These are the vocabulary words in this Concept.
- Slope
- the slant of a line or the steepness of a line. It is represented on a graph by a ratio of rise over run.
- Rise
- the vertical measurement of a line.
- Run
- the horizontal measurement of a line.
- Positive Slope
- a slope that goes up from left to right.
- Negative Slope
- a slope that goes down from right to left.
Guided Practice
Here is one for you to try on your own.
Is this slope positive, negative or undefined?
Answer
This line is vertical, therefore the slope of the line is undefined.
Video Review
Here is a video for review.
- This is a Khan Academy video on the slope of a line.
Practice
Directions: For each graph, tell if the slope of the line shown is positive, negative, zero, or undefined.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Directions: Answer each question.
11. Does a positive slope have to contain positive numbers?
12. True or false. A horizontal line is undefined.
13. True or false. A negative slope goes down from right to left.
14. True or false. A vertical line has an undefined slope.
15. True or false. You can figure out any slope as long as the line has some slant to it.
Horizontally
Horizontally means written across in rows.Negative Slope
A line with a negative slope will slant down from left to right.Positive Slope
A line with a positive slope will slant up from left to right.Rise
When calculating slope, the rise is the vertical distance between two specified points.Run
When calculating slope, the run is the horizontal distance traveled.Slope
Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the over the change in the .” The symbol for slope isVertically
Vertically means written up and down in columns.Image Attributions
Here you'll learn to recognize the slope of a line as the ratio of the vertical rise to the horizontal run and distinguish between types of slopes.
Concept Nodes:
Horizontally
Horizontally means written across in rows.Negative Slope
A line with a negative slope will slant down from left to right.Positive Slope
A line with a positive slope will slant up from left to right.Rise
When calculating slope, the rise is the vertical distance between two specified points.Run
When calculating slope, the run is the horizontal distance traveled.Slope
Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the over the change in the .” The symbol for slope isVertically
Vertically means written up and down in columns.